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# Independent derivations and source caveat for T602
The skill manifest for this run is
`agents/runs/20261029T055510Z-build-skills/manifest.json`, archived locally and
attached as skill-manifest.json. Both versions have SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
## Scale and elasticity convention
If -Delta u_D=1, u_{cD}(cx)=c^2*u_D(x), so T(cD)=c^4*T(D).
Love §224 has a stress function Psi with -Delta Psi=2 and torsional rigidity
C=2*mu*integral Psi; therefore Psi=2u, C=4*mu*T and the geometric engineering
constant J=C/mu is 4T. This fixes the factor for the OEIS constants. Love
§225(a,b) gives J=pi*a^4/2 and J=pi*a^3*b^3/(a^2+b^2).
## Polynomial solutions
The unit disk has u=(1-x^2-y^2)/4, whose integral is pi/8.
The ellipse with semiaxes (a,b) has
u=a^2*b^2*(1-x^2/a^2-y^2/b^2)/(2*(a^2+b^2)).
Its boundary values and Laplacian are exact polynomial identities. Pulling
back to the unit disk gives T=pi*a^3*b^3/(4*(a^2+b^2)).
For the equilateral triangle with the stated vertices,
u=y*(sqrt(3)*x-y)*(sqrt(3)*(1-x)-y)/(2*sqrt(3)).
Using y=sqrt(3)*z gives the rational polynomial
u=(3/2)*z*(x-z)*(1-x-z). The Laplacian becomes d_xx+(1/3)*d_zz.
Its negative is 1; all three boundary factors vanish. Integrating first
z<x<1-z, then 0<z<1/2, and multiplying by the Jacobian sqrt(3), gives
T=sqrt(3)/320. checks.py performs these identities exactly over QQ.
## Circular sectors
For 0<r<1 and 0<theta<alpha, expand 1 in the sine basis:
1=sum_{n odd}4/(n*pi)*sin(n*pi*theta/alpha).
Writing lambda=n*pi/alpha gives each radial coefficient
4*(r^2-r^lambda)/(n*pi*(lambda^2-4)). Integrating it gives
T=2*alpha/pi^2 * sum_{n odd}1/[n^2*(n*pi/alpha+2)^2].
For alpha=pi and 2*pi no odd n has lambda=2, so no resonant term occurs.
Set s=2*alpha/pi. The identity
1/[n^2*(n+s)^2]=(1/n^2+1/(n+s)^2)/s^2
-2*(1/n-1/(n+s))/s^3
reduces the odd sum to pi^2/(4*s^2)
-sum_{n odd, n<s}1/n^2/s^2 -2*sum_{n odd, n<s}1/n/s^3.
This proves T_half=pi/8-1/pi and T_slit=pi/4-16/(9*pi).
Both banks of the slit have zero boundary trace; the slit disk is not a disk.
## Rectangular and triangular spectral checks
On (0,r)x(0,1), the orthonormal modes are
2/sqrt(r)*sin(m*pi*x/r)*sin(n*pi*y), with squared integral
64*r/(pi^4*m^2*n^2) for odd m,n and zero otherwise.
Their eigenvalues are pi^2*((m/r)^2+n^2). Summing b_mn^2/lambda_mn
through m,n<=256 gives a lower bound. Parseval says the unsummed mass is
area minus the partial sum of b_mn^2. Every omitted eigenvalue is at least
pi^2*(257)^2/max(1,r^2), giving an upper bound on the omitted torsion.
The known integral of the square's first normalized sine mode is 8/pi^2;
this is the nonzero control in checks.py.
The triangle 0<y<x<1 is isometric to x,y>0, x+y<1. Its normalized modes
are 2*(sin(m*pi*x)*sin(n*pi*y)-sin(n*pi*x)*sin(m*pi*y)).
Integrating directly for m odd and n even gives
b_mn=-8*n/[pi^2*m*(m^2-n^2)]. Enumerating all such m,n lists each
unordered opposite-parity pair once. Same-parity pairs have integral zero.
The eigenvalue is pi^2*(m^2+n^2). The omitted mass is 1/2 minus the
partial squared-integral sum, and every omitted eigenvalue is at least
pi^2*257^2. This produces an independent enclosure containing the generator's
triangle result, without using a hyperbolic series.
**Source discrepancy resolved:** Comer–McDonald arXiv:2105.07477v1, PDF page 5,
states squared triangle projection integrals 16/(pi^4*j^2*k^2).
For {j,k}={1,2}, direct integration gives 256/(9*pi^4), whereas that expression
gives 4/pi^4. The stated eigenfunctions themselves integrate to the former.
This is a coefficient discrepancy, not a normalization choice. Accordingly
only Proposition 2.1 is used from that paper for production. The triangle
production formula is OEIS A180314 divided by four, and is independently
checked using the correctly integrated sine modes and the published decimal
digits. No unsupported selection between disagreeing computations is made.
## Precision evidence
The finite spectral checks are coarse independent checks, with explicit
Parseval remainder bounds, not 100-digit certification. The production
hyperbolic sums remove their algebraic zeta(5) tails exactly and enclose the
remaining exponential tails using geometric majorants. All rounding is Arb
ball arithmetic. 501 working bits and 80 odd terms are compared to 665 bits
and 120 odd terms. Written 100-significant-digit intervals must contain the
higher-precision enclosures. Polynomial and sector closed forms, aspect-ratio
reciprocity, normalization conversion, and the Saint-Venant inequality are
also checked on the values read back from the draft.